Kampyle of Eudoxus

Graph of Kampyle of Eudoxus

The Kampyle of Eudoxus (Greek: καμπύλη [γραμμή], meaning simply "curved [line], curve") is a curve, with a Cartesian equation of

$x^4=x^2+y^2$

from which the solution x = y = 0 should be excluded.

Alternative parameterizations

In polar coordinates, the Kampyle has the equation

$r= \sec^2\theta\,.$

Equivalently, it has a parametric representation as,

${\displaystyle x=a\sec(t),y=a\tan(t)\sec(t)}$.

History

This quartic curve was studied by the Greek astronomer and mathematician Eudoxus of Cnidus (c. 408 BC – c.347 BC) in relation to the classical problem of doubling the cube.

Properties

The Kampyle is symmetric about both the $x$- and $y$-axes. It crosses the $x$-axis at $(-1,0)$ and $(1,0)$. Besides these two points, it has no integer points. It has inflection points at

$(\pm\sqrt{3/2},\pm\sqrt{3}/2)$

(four inflections, one in each quadrant). The top half of the curve is asymptotic to $x^2-\frac12$ as $x\to \infty$, and in fact can be written as

$y = x^2\sqrt{1-x^{-2}} = x^2 - \frac12 \sum_{n \ge 0} C_n(2x)^{-2n}$

where

$C_n = \frac1{n+1} \binom{2n}{n}$

is the $n$th Catalan number.

See also

• List of curves

References

• J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 141–142. ISBN 0-486-60288-5.

External links

• O'Connor, John J.; Robertson, Edmund F., "Kampyle of Eudoxus", MacTutor History of Mathematics archive, University of St Andrews.
• Weisstein, Eric W. "Kampyle of Eudoxus". MathWorld.